Cluster Computing 5, 193–204, 2002
 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.
WCA: A Weighted Clustering Algorithm for Mobile
Ad Hoc Networks ∗
MAINAK CHATTERJEE, SAJAL K. DAS and DAMLA TURGUT
Center for Research in Wireless Mobility and Networking (CReWMaN), Department of Computer Science and Engineering,
University of Texas at Arlington, Arlington, TX 76019-0015, USA
Abstract. In this paper, we propose an on-demand distributed clustering algorithm for multi-hop packet radio networks. These types of
networks, also known as ad hoc networks, are dynamic in nature due to the mobility of nodes. The association and dissociation of nodes to
and from clusters perturb the stability of the network topology, and hence a reconfiguration of the system is often unavoidable. However, it
is vital to keep the topology stable as long as possible. The clusterheads, form a dominant set in the network, determine the topology and its
stability. The proposed weight-based distributed clustering algorithm takes into consideration the ideal degree, transmission power, mobility,
and battery power of mobile nodes. The time required to identify the clusterheads depends on the diameter of the underlying graph. We
try to keep the number of nodes in a cluster around a pre-defined threshold to facilitate the optimal operation of the medium access control
(MAC) protocol. The non-periodic procedure for clusterhead election is invoked on-demand, and is aimed to reduce the computation and
communication costs. The clusterheads, operating in “dual" power mode, connects the clusters which help in routing messages from a
node to any other node. We observe a trade-off between the uniformity of the load handled by the clusterheads and the connectivity of
the network. Simulation experiments are conducted to evaluate the performance of our algorithm in terms of the number of clusterheads,
reaffiliation frequency, and dominant set updates. Results show that our algorithm performs better than existing ones and is also tunable to
different kinds of network conditions.
Keywords: ad hoc networks, clusters, dominant set, load balancing
1. Introduction
Current wireless cellular networks solely rely on the wired
backbone by which all base stations are connected, implying
that networks are fixed and constrained to a geographical area
with a pre-defined boundary. Deployment of such networks
takes time and cannot be set up in times of utmost emergency.
Therefore, mobile multi-hop radio networks, also called ad
hoc or peer-to-peer networks, play a critical role in places
where a wired (central) backbone is neither available nor economical to build, such as law enforcement operations, battle
field communications, disaster recovery situations, and so on.
Such situations demand a network where all the nodes including the base stations are potentially mobile, and communication must be supported untethered between any two nodes.
A multi-cluster, multi-hop packet radio network architecture for wireless systems should be able to dynamically
adapt itself with the changing network configurations. Certain nodes, known as clusterheads, are responsible for the
formation of clusters each consisting of a number of nodes
(analogous to cells in a cellular network) and maintenance of
the topology of the network. The set of clusterheads is known
as a dominant set. A clusterhead does the resource allocation
to all the nodes belonging to its cluster. Due to the dynamic
nature of the mobile nodes, their association and dissociation
to and from clusters perturb the stability of the network and
∗ This work is partially supported by Texas Advanced Research Program
grant TARP-003594-013, Texas Telecommunications Engineering Consortium (TxTEC) and Nortel Networks.
thus reconfiguration of clusterheads is unavoidable. This is an
important issue since frequent clusterhead changes adversely
affect the performance of other protocols such as scheduling,
routing and resource allocation that rely on it. Choosing clusterheads optimally is an NP-hard problem [4]. Hence existing solutions to this problem are based on heuristic (mostly
greedy) approaches and none attempts to retain the stability
of the network topology [4,9]. We believe a good clustering scheme should preserve its structure as much as possible
when nodes are moving and/or the topology is slowly changing. Otherwise, re-computation of clusterheads and frequent
information exchange among the participating nodes will result in high computation overhead.
The concept of dividing the geographical region to be covered into small zones has been presented implicitly as clustering in the literature [13]. A natural way to map a “standard"
cellular architecture into a multi-hop packet radio network is
via the concept of a virtual cellular network [9]. Any node
can become a clusterhead if it has the necessary functionality, such as processing and transmission power. Nodes register with the nearest clusterhead and become members of that
cluster. Clusters may change dynamically, reflecting the mobility of the underlying network. The focus of the existing
literature in this area has mostly been on partitioning the network into clusters [4,5,11,17,18], without taking into consideration the efficient functioning of all the system components.
The lack of rigorous methodologies applicable to the design
and analysis of peer-to-peer mobile networks has motivated
in-depth research in this area. There have been solutions for
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efficient ways of interconnecting the nodes such that the latency of the system is minimized while throughput is maximized [11]. Most of the approaches [4,9,11] for finding the
clusterheads do not produce an optimal solution with respect
to battery usage, load balancing and MAC functionality.
In this paper, we propose a weight based distributed clustering algorithm which takes into consideration the number
of nodes a clusterhead can handle ideally (without any severe degradation in the performance), transmission power,
mobility, and battery power of the nodes. Unlike other existing schemes which are invoked periodically resulting in
high communication overhead, our algorithm is adaptively invoked based on the mobility of the nodes. More precisely, the
clusterhead election procedure is delayed as long as possible to reduce the computation cost. Balancing the loads between clusterheads is another desirable feature of any clustering algorithm, however, it is very difficult to maintain a
completely balanced system due to the dynamic nature of the
nodes. Our algorithm achieves load balancing by specifying
a pre-defined threshold on the number of nodes that a clusterhead can handle ideally. This ensures that none of the clusterheads are overloaded at any instance of time. We define
load balancing factor (LBF) to measure the degree of load
balancing among the clusterheads. Connecting the nodes is
another important issue since the nodes need to communicate with each other. In order to consider any kind of routing
between the clusters, it is essential that the clusters are connected and the nodes are able to route messages via the clusterheads. We define connectivity as the probability that a node
is reachable from any other node. Clusterheads in our scheme
work in dual power mode. The clusterheads can operate at a
higher power mode (resulting in a higher transmission range)
for inter-cluster communication while they use lower power
for intra-cluster communication. Finally, we conduct detailed
simulation experiments and demonstrate that our clustering
algorithm yields better results as compared to the existing
heuristics in terms of the number of reaffiliations (detachment
of a node from its current cluster and attachment to another
existing cluster) and dominant set updates.
The rest of the paper is organized as follows. In section 2, we summarize previous work and their limitations.
In section 3, we propose the Weighted Clustering Algorithm
(WCA). Simulation results are presented in section 4 while
conclusions are offered in section 5.
2. Previous work
Several heuristics have been proposed to choose clusterheads
in ad hoc networks. They include (i) Highest-Degree heuristic, (ii) Lowest-ID heuristic, and (iii) Node-Weight heuristic.
The Lowest-ID and the Highest-Degree were the two clustering algorithms which were based on the link-cluster architecture [2,3,10]. In the assumed graph model of the network,
the mobile terminals are represented as nodes; there exists
an edge between two nodes if they can communicate with
each other directly (i.e., one node lies within the transmission range of another). The performance of the above three
CHATTERJEE, DAS AND TURGUT
heuristics were studied in [5,11] by simulation experiments
in which mobile nodes were randomly placed in a square grid
and moved with different speeds in different directions.
2.1. Highest-Degree heuristic
The Highest-Degree, also known as connectivity-based clustering, was originally proposed by Gerla and Parekh [11,20]
in which the degree of a node is computed based on its distance from others. Each node broadcasts its id to the nodes
that are within its transmission range. A node x is considered to be a neighbor of another node y if x lies within the
transmission range of y. The node with maximum number of
neighbors (i.e., maximum degree) is chosen as a clusterhead
and any tie is broken by the unique node ids. The neighbors
of a clusterhead become members of that cluster and can no
longer participate in the election process. Since no clusterheads are directly linked, only one clusterhead is allowed per
cluster. Any two nodes in a cluster are at most two-hops away
since the clusterhead is directly linked to each of its neighbors
in the cluster. Basically, each node either becomes a clusterhead or remains an ordinary node (neighbor of a clusterhead).
Experiments demonstrate that the system has a low rate
of clusterhead change but the throughput is low under the
Highest-Degree heuristic. Typically, each cluster is assigned
some resources which is shared among the members of that
cluster on a round-robin basis [11,17,18]. As the number of
nodes in a cluster is increased, the throughput drops and hence
a gradual degradation in the system performance is observed.
The reaffiliation count of nodes are high due to node movements and as a result, the highest-degree node (the current
clusterhead) may not be re-elected to be a clusterhead even
if it looses one neighbor. All these drawbacks occur because
this approach does not have any restriction on the upper bound
on the number of nodes in a cluster.
2.2. Lowest-ID heuristic
The Lowest-ID, also as known as identifier-based clustering,
was originally proposed by Baker and Ephremides [2,3,10].
This heuristic assigns a unique id to each node and chooses
the node with the minimum id as a clusterhead. Thus, the
ids of the neighbors of the clusterhead will be higher than
that of the clusterhead. However, the clusterhead can delegate its responsibility to the next node with the minimum
id in its cluster. A node is called a gateway if it lies within
the transmission range of two or more clusterheads. Gateway
nodes are generally used for routing between clusters. Only
gateway nodes can listen to the different nodes of the overlapping clusters that they lie. The concept of distributed gateway
(DG) is also used for inter-cluster communication only when
the clusters are not overlapping. DG is a pair of nodes that lies
in different clusters but they are within the transmission range
of each other. The main advantage of distributed gateway is
maintaining connectivity in situations where any clustering
algorithm fails to provide connectivity.
WEIGHTED CLUSTERING ALGORITHM
For this heuristic, the system performance is better compared with the Highest-Degree heuristic in terms of throughput. Since the environment under consideration is mobile, it is
unlikely that node degrees remain stable resulting in frequent
clusterhead updates. However, the drawback of this heuristic
is its bias towards nodes with smaller ids which may lead to
the battery drainage of certain nodes. One might think that
this problem may be fixed by renumbering the node ids from
time to time, which is however non-trivial. There are other
problems associated with such renumbering. For instance,
the optimal frequency of renumbering would need to be determined so that the system performance is maximized. More
importantly, every time node ids are reshuffled, the neighboring list of all the nodes need also to be changed. If we consider that the nodes are numbered in the increasing order of
their remaining battery power, then a centralized algorithm is
required. We can avoid this by exchanging ids between nodes
and making sure that the uniqueness of ids are maintained.
Even then, the clustering has to be redone which would add
unnecessary computational complexity to the system. For
example, suppose two nodes mutually exchange their ids in
order to keep the ids according to their remaining battery
power. After exchanging, all nodes that were connected to
these two nodes, regardless of their status (clusterhead or ordinary node), need to be notified of the change so that they
can update their neighbor list. This effect may propagate and
add overhead to the system. Moreover, it does not attempt to
balance the load uniformly across all the nodes.
2.3. Node-Weight heuristic
Basagni et al. [5,6] proposed two algorithms, namely distributed clustering algorithm (DCA) and distributed mobilityadaptive clustering algorithm (DMAC). In this approach,
each node is assigned weights (a real number 0) based on
its suitability of being a clusterhead. A node is chosen to be
a clusterhead if its weight is higher than any of its neighbor’s
weight; otherwise, it joins a neighboring clusterhead. The
smaller node id is chosen in case of a tie. The DCA makes an
assumption that the network topology does not change during the execution of the algorithm. Thus, it is proven to be
useful for “quasi-static” networks when the nodes either do
not move or move very slowly. The other assumptions are:
(i) the messages are guaranteed to be delivered to all of the
nodes’ neighbors within a finite amount of time, and (ii) every
node is aware of the ids and the corresponding weights of all
the nodes which are only one hop away. The DMAC algorithm, on the other hand, adapts itself to the network topology
changes and therefore can be used for any mobile networks.
To verify the performance of the system, the nodes were
assigned weights which varied linearly with their speeds
but with negative slope. Results proved that the number
of updates required is smaller than the Highest-Degree and
Lowest-ID heuristics. Since node weights were varied in each
simulation cycle, computing the clusterheads becomes very
expensive and there are no optimizations on the system parameters such as throughput and power control.
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2.4. Limitations of existing heuristics
None of the above three heuristics leads to an optimal election
of clusterheads since each deals with only a subset of parameters which can possibly impose constraints on the system.
Each of these heuristics is suitable for a specific application
rather than for arbitrary wireless mobile networks.
To be precise, the Highest-Degree heuristic states that the
node with the largest number neighbors should be elected as
a clusterhead. However, a clusterhead may not be able handle
a large number of nodes due to resource limitations even if
these nodes are its immediate neighbors and lie well within
its transmission range. For example, Bluetooth [12] employs
a Master-Slave model where the clusterhead is the master and
can handle up to seven slaves [21]. Thus, the load handling
capacity of the clusterhead puts an upper bound on the nodedegree. In other words, simply covering the area with the
minimum number of clusterheads will put more burden on
the clusterheads. On the other hand, a large number of clusterheads will lead to a computationally expensive system. Although this may result in good throughput, the data packets
have to go through multiple hops thus implying high latency.
Similarly, the Lowest-ID heuristic concerns only with the
lowest node ids which are arbitrarily assigned numbers without considering the qualifications of a node possibly being
elected as a clusterhead. Since the node ids do not change
with time, those with smaller ids are more likely to become
clusterheads than nodes with larger ids. Thus, certain nodes
are prone to power drainage due to serving as clusterheads for
longer periods of time.
The Node-Weight heuristic assigns node-weights based on
the suitability of nodes acting as clusterheads and the election
of the clusterhead is done on the basis of the largest weight
among its neighbors. This means that a node decides to become a clusterhead or stay as an ordinary node depending on
the weights of its one hop neighbors. Basically, the node has
to wait for all the responses from its neighbors to make its
own decision to be a clusterhead or an ordinary node. This
heuristic does not account for the amount of time that a node
may need to wait to receive responses from its neighbors.
3. Weighted Clustering Algorithm (WCA)
In this section, we present our weighted clustering algorithm.
We give the design philosophy and the basis of our algorithm
before discussing the details.
3.1. Preliminaries
The network formed by the nodes and the links can be represented by an undirected graph G = (V , E), where V represents the set of nodes vi and E represents the set of links ei .
Note that the cardinality of V remains the same but the cardinality of E always changes with the creation and deletion of
links. Clustering can be thought as a graph partitioning problem with some added constraints. As the underlying graph
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CHATTERJEE, DAS AND TURGUT
does not show any regular structure, partitioning the graph
optimally (i.e., with minimum number of partitions) with respect to certain parameters becomes an NP-hard problem [7].
More formally, we look for the set of vertices S ⊆ V (G),
such that
N[v] = V (G).
v∈S
Here, N[v] is the neighborhood of node v, defined as
N[v] =
v | dist(v, v ) < txrange ,
v ∈V , v =v
where txrange is the transmission range of v. The neighborhood of a clusterhead is the set of nodes which lie within its
transmission range. The set S is called a dominating set such
that every vertex of G belongs to S or has a neighbor in S.
The dominating set of the graph is the set of clusterheads.
It might be possible that a node is physically nearer to a clusterhead but belongs to another clusterhead. This is because of
the other considerations discussed in section 2.4. For example, a node might be physically closer to a clusterhead that is
over loaded. In that case it will attach itself to a clusterhead
which is far off due to mobility, the nodes may go outside
the transmission range of their clusterhead thus changing its
neighborhood. However, this does not result in a change of
the dominant set. It might so happen that the detached node
is not able to attach itself to any of the nodes in the dominant
set. This implies that the existing dominant set can no longer
cover the entire graph, hence the clustering algorithm has to
be invoked to find a new dominant set.
3.2. Design philosophy
Choosing an optimal number of clusterheads which will yield
high throughput but incur as low latency as possible, is still
an important problem. As the search for better heuristics for
this problem continues, we propose a new algorithm which
is based on the use of a combined weight metric, that takes
into account several system parameters like the ideal nodedegree, transmission power, mobility and the battery power
of the nodes. Depending on specific applications, any or all
of these parameters can be used in the metric to elect the clusterheads. We could have a fully distributed system where all
the nodes in the mobile network share the same responsibility and act as clusterheads. However, more clusterheads result in extra number of hops for a packet when it gets routed
from the source to the destination, since the packet has to go
via larger number of clusterheads. Thus this solution leads to
higher latency, more power consumption and more information processing per node.
On the other hand, to maximize the resource utilization,
we can choose to have the minimum number of clusterheads
to cover the whole geographical area over which the nodes
are distributed. The whole area can be split up into zones, the
size of which can be determined by the transmission range
of the nodes. This can put a lower bound on the number
of clusterheads required. Ideally, to reach this lower bound,
a uniform distribution of the nodes is necessary over the entire area. Also, the total number of nodes per unit area should
be restricted so that the clusterhead in a zone can handle all
the nodes therein. However, the zone based clustering is not a
viable solution due to the following reasons. The clusterheads
would typically be centrally located in the zone, and if they
move, new clusterheads have to be elected. It might so happen
that none of the other nodes in that zone are centrally located.
Therefore, to find a new node which can act as a clusterhead
with the other nodes within its transmission range might be
difficult. Another problem arises due to non-uniform distribution of the nodes over the whole area. If a certain zone
becomes densely populated due to migration of nodes from
other zones, then the clusterhead might not be able to handle all the traffic generated by the nodes because there is an
inherent limitation on the number of nodes a clusterhead can
handle. We propose to elect the minimum number of clusterheads which can support all the nodes in the system satisfying
the above constraints.
3.3. Basis for our algorithm
To decide how well suited a node is for being a clusterhead,
we take into account its degree, transmission power, mobility
and battery power. The following features are considered in
our clustering algorithm:
• The clusterhead election procedure is not periodic and is
invoked as rarely as possible. This reduces system updates
and hence computation and communication costs. The
clustering algorithm is not invoked if the relative distances
between the nodes and their clusterheads do not change.
• Each clusterhead can ideally support only δ (a pre-defined
threshold) nodes to ensure efficient medium access control
(MAC) functioning. If the clusterhead tries to serve more
nodes than it is capable of, the system efficiency suffers
in the sense that the nodes will incur more delay because
they have to wait longer for their turn (as in TDMA) to get
their share of the resource. A high system throughput can
be achieved by limiting or optimizing the degree of each
clusterhead.
• The battery power can be efficiently used within certain
transmission range, i.e., it will take less power for a node
to communicate with other nodes if they are within close
distance to each other. A clusterhead consumes more battery power than an ordinary node since a clusterhead has
extra responsibilities to carry out for its members.
• Mobility is an important factor in deciding the clusterheads. In order to avoid frequent clusterhead changes, it
is desirable to elect a clusterhead that does not move very
quickly. When the clusterhead moves fast, the nodes may
be detached from the clusterhead and as a result, a reaffiliation occurs. Reaffiliation takes place when one of the
ordinary nodes moves out of a cluster and joins another
existing cluster. In this case, the amount of information
exchange between the node and the corresponding clusterhead is local and relatively small. The information update
WEIGHTED CLUSTERING ALGORITHM
in the event of a change in the dominant set is much more
than a reaffiliation.
• A clusterhead is able to communicate better with its neighbors having closer distances from it within the transmission range [22]. As the nodes move away from the clusterhead, the communication may become difficult due mainly
to signal attenuation with increasing distance.
3.4. Proposed algorithm
Based on the preceding discussion, we propose an algorithm
called weighted clustering algorithm (WCA) that effectively
combines each of the above system parameters with certain
weighing factors chosen according to the system needs. For
example, power control is very important in CDMA networks,
thus the weight of the corresponding parameter can be made
larger. The flexibility of changing the weight factors helps us
apply our algorithm to various networks. The output of clusterhead election procedure is a set of nodes called the dominant set. According to our notation, the number of nodes
that a clusterhead can handle ideally is δ. This is to ensure
that clusterheads are not over-loaded and the efficiency of the
system is maintained at the expected level. The clusterhead
election procedure is invoked at the time of system activation
and also when the current dominant set is unable to cover all
the nodes. Every invocation of the election algorithm does
not necessarily mean that all the clusterheads in the previous dominant set are replaced with the new ones. If a node
detaches itself from its current clusterhead and attaches to another clusterhead then the involved clusterheads update their
member list instead of invoking the election algorithm. A preliminary version of this algorithm appeared in [8].
3.4.1. Clusterhead election procedure
The procedure consists of eight steps as described below:
Step 1. Find the neighbors of each node v (i.e., nodes within
its transmission range) which defines its degree, dv , as
dv = N(v) =
dist(v, v ) < txrange .
v ∈V , v =v
Step 2. Compute the degree-difference, v = |dv − δ|, for
every node v.
Step 3. For every node, compute the sum of the distances,
Dv , with all its neighbors, as
Dv =
dist(v, v ) .
v ∈N(v)
Step 4. Compute the running average of the speed for every
node till current time T . This gives a measure of mobility
and is denoted by Mv , as
Mv =
T 1 (Xt − Xt −1 )2 + (Yt − Yt −1 )2 ,
T
t =1
where (Xt , Yt ) and (Xt −1 , Yt −1 ) are the coordinates of the
node v at time t and (t − 1), respectively.
197
Step 5. Compute the cumulative time, Pv , during which a
node v acts as a clusterhead. Pv implies how much battery power has been consumed which is assumed more for
a clusterhead than an ordinary node.
Step 6. Calculate the combined weight Wv for each node v,
where
Wv = w1 v + w2 Dv + w3 Mv + w4 Pv ,
where w1 , w2 , w3 and w4 are the weighing factors for the
corresponding system parameters.
Step 7. Choose that node with the smallest Wv as the clusterhead. All the neighbors of the chosen clusterhead are no
longer allowed to participate in the election procedure.
Step 8. Repeat steps 2–7 for the remaining nodes not yet selected as a clusterhead or assigned to a cluster.
The first component, w1 v , contributing towards the combined metric Wv helps in efficient MAC functioning because
it is always desirable for a clusterhead to handle upto a certain
number of nodes in its cluster. The motivation of Dv is mainly
related to energy consumption. It is known that more power is
required to communicate to a larger distance. As a result, one
might think that it would be more appropriate to use the sum
of the squares (or higher exponent) of the distances, because
the power required to support a link increases considerably
faster than linearly with distance (at least in the far-field region). The usual attenuation in the signal strength is inversely
proportional to some exponent of the distance, which is usually approximated to 4 in cellular networks where the distance
between mobiles and base stations is of the order of 2–3 miles.
But in ad hoc networks, the distances involved are rather small
(approximately hundreds of meters). In this range, the attenuation can be assumed to be linear [16]. The third component
for Wv is due to mobility of the nodes. As discussed in section 3.3, a node with less mobility is always a better choice
for a clusterhead. The last component Pv , is measured as the
total (cumulative) time a node acts as a clusterhead. We have
assumed that the battery power of all nodes to be the same at
the beginning. In that case, the battery drainage gives a direct
measure of the available battery power. Also, we have taken
into consideration that the battery drainage will be more for
nodes acting as clusterheads. However, if the nodes have various battery power to start with, then it would be a more accurate metric to measure the power currently available at the
node. This will in turn depends on the node’s initial power
and the power expended based on actual network traffic and
length of the links used to support it.
3.4.2. An illustrative example
We demonstrate our weighted clustering algorithm with the
help of figures 1–6. All numeric values, as obtained from executing WCA on the 15 nodes as shown in figure 1, are tabulated in table 1. Figure 1 shows the initial configuration of the
nodes in the network with individual node ids. Dotted circles
with equal radius represent the fixed transmission range for
each node. A node can hear broadcast beacons from the nodes
which are within its transmission range. An edge between two
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CHATTERJEE, DAS AND TURGUT
Figure 5. Clusters identified.
Figure 1. Initial configuration of nodes.
Figure 6. Connectivity achieved.
Table 1
Execution of WCA.
Figure 2. Neighbors identified.
Figure 3. Velocity of the nodes.
Figure 4. Clusterheads identified.
Node
id
dv
step 1
v
step 2
Dv
step 3
Mv
step 4
Pv
step 5
Wv
step 6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
2
1
1
1
3
1
2
2
4
3
1
2
2
2
1
0
1
1
1
1
1
0
0
2
1
1
0
0
0
1
6
4
3
3
9
3
6
7
13
12
3
5
7
5
3
2
2
3
4
1
2
0
3
2
2
0
3
3
2
4
1
2
1
2
4
2
0
3
6
7
1
4
2
0
3
1.35
1.70
1.50
1.60
2.75
1.50
1.20
1.70
4.40
3.55
1.35
1.35
1.65
1.10
1.65
nodes in figure 2 signifies that the nodes are neighbors of each
other. The degree, dv , which is the total number of neighbors
a node has is shown in step 1. The degree difference, v ,
of each node with ideal node degree δ = 2 is computed in
step 2. Sum of the distances, Dv , for each node is calculated
as step 3, where the unit distance has been chosen arbitrarily.
The arrows in figure 3 represent the speed and direction of
movement associated with every node. A longer arrow represents faster movement and a shorter arrow indicates slower
movement. The values for Mv (step 4), are chosen randomly.
Mv = 0 implies that a node does not move at all. We choose
WEIGHTED CLUSTERING ALGORITHM
some arbitrary values for Pv which represent the amount of
time a node has acted as a clusterhead. This corresponds to
step 5 in our algorithm. After the values of all the components
are identified, we compute the weighted metric, Wv , for every
node as proposed in step 6 in our algorithm. The weights considered are w1 = 0.7, w2 = 0.2, w3 = 0.05 and w4 = 0.05.
Note that these weighing factors are chosen arbitrarily such
that w1 + w2 + w3 + w4 = 1. The contribution of the individual components can be tuned by choosing the appropriate
combination of the weighing factors. Figure 4 shows how a
node with minimum Wv is selected as the clusterhead in a distributed fashion as stated in step 7 in our algorithm. The solid
nodes represent the clusterheads elected for the network. Note
that no two clusterheads are immediate neighbors. Figure 5
shows the initial clusters formed by execution of the clustering algorithm. We observe that the total number of neighbors
served by each clusterhead is close to the predefined ideal degree, δ = 2. Figure 6 shows the achieved connectivity in
the network. As discussed earlier, the connectivity is accomplished through the higher power (as a result of dual mode
power) transmission range of a clusterhead. It can be noted
that a single component graph is obtained in this case which
means that there is a path from a node to any other node.
3.4.3. Complexity due to distributiveness
The time required for the selection of the node with minimum Wv depends on the implementation of the algorithm.
In a centralized system with a central server, the minimum
Wv can be found in linear time with respect to the number of
nodes. But it is not possible to have a centralized server in
ad hoc networks. So, we proceed with a distributed solution
in which all the nodes broadcast their ids along with Wv values. A node receives broadcasts from its neighbors and stores
the information. This stored information is again exchanged
with the immediate neighbors and the process continues till
all the nodes become aware of the node with the smallest Wv .
The time required for the nodes to gather information about
all other nodes will depend on the diameter of the underlying
graph. It is to be noted that this procedure yields the global
minima of Wv s unlike the Lowest-ID algorithm which finds
only the local minima of ids.
It can be argued that the existing heuristics discussed in
section 2 are all special cases of our algorithm. The HighestDegree heuristic considers only the degree of a node and disregards all other system parameters (w2 = w3 = w4 = 0).
In Lowest-ID heuristic, the assignment of the ids are random. We can assume that the ids being assigned are based
on mobility. The lowest id is assigned to the least mobile node and highest id for the most mobile. In that case,
(w1 = w2 = w4 = 0). The Node-Weight heuristic simply
assigns weights to the nodes which are equivalent to Wv in
our case. The basis for suitability of nodes being clusterheads
is ignored there. However, in our approach, we define and
formulate the parameters for choosing a clusterhead and we
show how the weight Wv are calculated.
199
3.4.4. System activation and update policy
When a system is initially brought up, every node v broadcasts its id which is registered by all other nodes lying within
v’s transmission range, tx_range, as can be seen from figure 1.
It is assumed that a node receiving a broadcast from another
node can estimate their mutual distance from the strength of
the signal received. GPS (Global Positioning System) can
be another solution since it is mainly used to obtain the geographical location of nodes. Even though GPS might make
the problem relatively simpler, but there is always a cost associated with the deployment of GPS since every mobile node
must be a GPS receiver. Based on the received signal strength,
every node is made aware of its neighboring nodes and their
corresponding distances. Note that these neighboring nodes
are only the geographical neighbors and do not necessarily
mean neighbors within the same cluster. Once the neighbors
list for each node is ready, our clustering algorithm chooses
the clusterhead for the first time, as illustrated in figure 4. It
can be noted that the mobility factor and the battery power
would be the same for all the nodes when the system is initialized. Effectively, Wv will have only two terms v and Dv
contributing to it. Each node maintains its status (i.e., clusterhead or not). A non-clusterhead node knows the cluster it
belongs to and the corresponding clusterhead.
Due to the dynamic nature of the system considered, the
nodes as well as the clusterheads tend to move in different
directions, thus disorganizing the stability of the configured
system. So, the system has to be updated from time to time.
The update may result in formation of new clusters and possible change of point of attachment of nodes from one clusterhead to another within the existing dominant set. This is
called reaffiliation. The frequency of update and hence reaffiliation is an important issue. If the system is updated periodically at a high frequency, then the latest topology of the
system can be used to find the clusterheads which will yield
a good dominant set. However, this will lead to high computational cost resulting in the loss of battery power or energy.
If the frequency of update is low, there are chances that current topological information will be lost resulting in sessions
terminated midway.
All the nodes continuously monitor their signal strength
as received from the clusterhead. When the mutual separation between the node and its clusterhead increases, the signal
strength decreases. In that case, the mobile has to notify its
current clusterhead that it is no longer able to attach itself to
that clusterhead. The clusterhead tries to hand-over the node
to a neighboring cluster (existing clusterhead in the dominant set). The clusterhead of the reaffiliated node updates
its member list. If the node goes into a region not covered
by any clusterhead, then the clusterhead election algorithm is
invoked and the new dominant set is obtained.
The objective of our clusterhead election algorithm is to
minimize the number of changes in dominant set update.
Once the neighbors list for all nodes are created, the degreedifference v is calculated for each node v. Also, Dv is computed for each node by summing up the distances of its neighbors. The mobility Mv is calculated by averaging the speed
200
of the node. The total amount of time, Tv , it remained as a
clusterhead is also calculated. All these parameters are normalized, which means that their values are made to lie in a
pre-defined region. The corresponding weights w1 , w2 , w3
or w4 are kept fixed for a given system. The weighing factors also give the flexibility of adjusting the effective contribution of each of the parameters in calculating the combined
weight Wv . For example, in a system where battery power
is more important, the weight w4 associated with Tv can be
made larger. Note that the sum of these weighing factors is 1.
The node with the minimum total weight, Wv , is elected as
a clusterhead. The elected clusterhead and its neighbors are
no longer eligible to participate in the remaining part of the
election process which continues until every node is found to
be either a clusterhead or a neighbor of some clusterhead.
CHATTERJEE, DAS AND TURGUT
We define connectivity as the probability that a node is
reachable from any other node. For a single component graph,
any node is reachable from any other node and the connectivity is 1. If the network does not result in a single component
graph, then we can say that all the nodes in the largest component can communicate with each other and the connectivity
can be the ratio of the cardinality of the largest component to
the cardinality of the graph. Thus,
connectivity =
size of largest component
.
N
The transmission range of a clusterhead can be made large
enough by adjusting the power in such a way so as to yield a
connected network.
3.7. Routing messages
3.5. Load balancing
The load handled by a clusterhead depends on the number of
nodes supported by it. A clusterhead, apart from supporting
its members with the radio resources, has also to route messages for other nodes belonging to different clusters. Therefore, it is not desirable to have any clusterhead overly loaded
while some others are lightly loaded [1]. At the same time, it
is difficult to maintain a perfectly load balanced system at all
times due to frequent detachment and attachment of the nodes
from and to the clusterheads. To quantitatively measure how
well balanced the clusterheads are, we introduce a parameter
called load balancing factor (LBF). As the load of a clusterhead can be represented by the cardinality of its cluster size,
the variance of the cardinalities will signify the load distribution. We define the LBF as the inverse of the variance of the
cardinality of the clusters. Thus,
nc
,
LBF = 2
i (xi − µ)
where nc is the number of clusterheads, xi is the cardinality
of cluster i, and µ = (N − nc )/nc , (N being the total number
of nodes in the system) is the average number of neighbors
of a clusterhead. Clearly, a higher value of LBF signifies a
better load distribution and it tends to infinity for a perfectly
balanced system.
3.6. Connecting the clusters
As a logical extension to clustering, we investigate the connectivity of the nodes which is essential for any routing algorithm. Clustering ensures that the nodes within a cluster
are able to communicate among themselves through the clusterheads, each of which acts as the central node of a star, as
shown in figure 5. But, inter-cluster communication is not
possible if the clusters are not connected. For two clusters
to communicate with each other, we assume that the clusterheads are capable of operating in dual power mode. A clusterhead uses low power to communicate with the members in its
transmission range, and high power to communicate with the
neighboring clusterheads because of greater range. The links
between the clusterheads are shown as solid lines in figure 6.
As our clusterhead connecting technique assures that all
nodes are connected with probability almost 1, we can route
messages from any node to any other node. Several algorithms have been proposed for routing messages in ad hoc
networks [13–15,19,20]. This paper does not propose or deal
with any routing algorithm. It can be noted that the number
of hops a message makes in a clusterhead based routing [19]
depends on the number of clusterheads in the network. It is
not recommended to have too few or too many clusterheads
in the network. It is our belief that our clustering algorithm
will help routing algorithms in terms of the number of hops.
at the clusterhead is expected to be minimum due to the load
distribution. If a source node A, wishes to establish a connection with a destination node B, then it first needs to discover
a route to B. Node A sends a “route discovery” request message containing B’s id to its clusterhead. If B is not present
in the same cluster as A, then A’s clusterhead propagates the
request message to its neighboring clusterhead. On receiving
the request, the clusterheads can check their member list for
B. This query is done in parallel. If B is found, then a positive acknowledgement is sent from B which reaches A via the
clusterheads and the route discovery procedure is terminated.
If B is not found then the request message is propagated to the
two-hop neighbors of A’s clusterhead and the process continues till B is found.
The worst-case search time will arise in the case where all
the clusterheads are such connected as to form linear graph.
Here, searching a cluster has to be done one at a time, eradicating the possibility of parallel search among the clusters.
The worst-case time complexity of finding a node will be
O(|clusterheads|), where |clusterheads| is the cardinality of
the dominant set.
4. Simulation study
We simulate a system of N nodes on a 100 × 100 grid.
The nodes could move in all possible directions with displacement varying uniformly between 0 to a maximum value
(max_disp), per unit time. To measure the performance of our
WEIGHTED CLUSTERING ALGORITHM
201
Figure 7. Average number of clusters, max_disp = 5.
Figure 8. Reaffiliations per unit time, max_disp = 5.
algorithm WCA, we identify three metrics: (i) the number
of clusterheads, (ii) the number of reaffiliations, and (iii) the
number of dominant set updates. Every time a dominant set
is identified, its cardinality gives the number of clusterheads.
The reaffiliation count is incremented when a node gets dissociated from its clusterhead and becomes a member of another
cluster within the current dominant set. The dominant set update takes place when a node can no longer be a neighbor of
any of the existing clusterheads. These three parameters are
studied for varying number of nodes (N) in the system, transmission range and maximum displacement. We also study
how the load balance factor changes as the system evolves
and how well connected the nodes are.
In our simulation experiments, N was varied between 20
and 60, and the transmission range was varied between 0 and
70. The nodes moved randomly in all possible directions
with a maximum displacement of 10 along each of the coordinates. At every time unit, the nodes move a distance that
is uniformly distributed between 0 and max_disp.√ Thus, the
maximum Euclidean displacement possible is 10 2. We assume that each clusterhead can handle at most δ = 10 nodes
(ideal degree) in its cluster in terms of resource allocation.
Due to the importance of keeping the node degree as close to
the ideal as possible, the weight w1 associated with v was
chosen high. The next higher weight w2 was given to Dv ,
which is the sum of distances. Mobility and battery power
were given low weights. The values used for simulation were
w1 = 0.7, w2 = 0.2, w3 = 0.05 and w4 = 0.05. Note that
these values are arbitrary at this time and should be adjusted
according to the system requirements.
Figure 9. Dominant set updates, max_disp = 5.
4.1. Experimental results
Figure 7 shows the variation of the average number of clusterheads with respect to the transmission range where max_disp
of 5. The results are shown for varying N. We observe that the
average number of clusterheads decreases with the increase in
the transmission range. This is due to the fact that a clusterhead with a large transmission range will cover a larger area.
Figure 8 shows the reaffiliations per unit time. For low transmission range, the nodes in a cluster are relatively close to
the clusterhead, and a detachment is unlikely. The number
of reaffiliations increases as the transmission range increases,
and reaches a peak when transmission range is between 25
and 30. Further increase in the transmission range results in a
decrease in the reaffiliations since the nodes, in spite of their
random motion, tend to stay inside the large area covered by
the clusterhead. Figure 9 shows the number of dominant set
updates with respect to the transmission range. For smaller
transmission range, the cluster area is small and the probability of a node moving out of its cluster is high. As the transmission range increases, the number of dominant set updates
decreases because the nodes stay within their cluster in spite
of their movements.
Figures 10–12 show the variation of the same parameters
but for varying max_disp and constant transmission range
of 30. Figure 10 shows that the average number of clusterheads is almost the same for different values of max_disp,
particularly for larger values of N. This is because, no matter what the mobility is, it simply results in a different configuration but the cluster size remains the same. Figure 11
shows the reaffiliations per unit time with respect to the maximum displacement. As the displacement becomes larger, the
nodes tend to move farther from their clusterhead, detaching
themselves from the clusterhead and causing more reaffilia-
202
CHATTERJEE, DAS AND TURGUT
Figure 10. Average number of clusters, tx_range = 30.
Figure 13. Load distribution.
Figure 11. Reaffiliations per unit time, tx_range = 30.
Figure 14. Connectivity.
Figure 12. Dominant set updates, tx_range = 30.
tions per unit time and more dominant set updates. These are
shown in figures 11 and 12, respectively.
The non-periodic invocation of our clustering algorithm
can be observed from figure 13 and the reachability of one
node from another is shown in figure 14. We observe that
after every dominant set update, there is a gradual increase
in the Load Balance Factor (LBF). This gradual increase in
LBF is due to the diffusion of the nodes among clusters. This
improvement does not increase indefinitely because the nodes
tend to move away from all possible clusterheads and the clustering algorithm has to be invoked to ensure connectivity. The
clustering algorithm tries to connect all the nodes at the cost
of load imbalance which is represented by the sharp decrease
in LBF. To study the reachability of one node from another, it
is essential that the clusters are connected. For this purpose,
the clusterheads operate in dual power modes. As mentioned
earlier, the lower power is used to communicate within the
cluster whereas the higher power (transmission range) is used
to communicate with the neighboring clusterheads. To obtain
the higher transmission range, we scaled up the lower transmission range by a constant factor. Simulation was conducted
for N = 50 and the constant factor was varied from 1.0 to 2.0
with increments of 0.25. Figure 14 demonstrates that a well
connected graph can be obtained at the cost of a higher transmission range.
Figure 15 shows the relative performance of the HighestDegree, Lowest-ID, Node-Weight heuristics and WCA in
terms of the number of reaffiliations per unit time vs. transmission range where N = 30. The number of reaffiliations for WCA is at most half the number obtained from the
Lowest-ID. The main reason is that the frequency of invoking the clustering algorithm is lower in WCA, thus resulting
WEIGHTED CLUSTERING ALGORITHM
203
Acknowledgements
The authors are grateful to the anonymous referees and the
guest editors for valuable suggestions which improved the
quality of the paper.
References
Figure 15. Comparison of reaffiliations, N = 30.
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the weighing factors according to the system needs.
5. Conclusions
We proposed a weight based distributed clustering algorithm
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Mainak Chatterjee received his B.Sc. degree in
physics (Hons) from the University of Calcutta in
1994. In 1998, he finished his M.E. in electrical communication engineering at the Indian Institute of Science, Bangalore. He is currently a Ph.D. candidate
in the Department of Computer Science and Engineering at the University of Texas at Arlington. He
has been recognized as a university scholar and has
been included in the WHO’S WHO among students
in American Universities and Colleges in 2001. His
research interests include mobile computing, MAC
layer protocols, CDMA data networking, multimedia communications and ad hoc networks. He has
published over a dozen research papers and is currently working on several industry funded projects.
He is a recipient of the TxTEC (Texas Telecommunications Engineering Consortium) fellowship and is
a IEEE-CS student member.
E-mail: [email protected]
Sajal K. Das is currently a Full Professor of Computer Science and Engineering and the founding Director of the Center for Research in Wireless Mobility and Networking (CReWMaN) at the University
of Texas at Arlington (UTA). During 1988–1999, he
was a Professor of Computer Science at the University of North Texas, where he twice (1991 and 1997)
received the Student Association Honor Professor
Award for best teaching and outstanding research, as
well as the Developing Scholar’s award in 1996 for
research excellence. He also received the Oustanding
Senior Faculty Research Award in computer science
and engineering at UTA. His current interests include
mobile computing, wireless networks, resource and
mobility management, QoS provisioning in wireless
multimedia, mobile Internet, and distributed computing. He has published over 200 research papers in
these areas and directed numerous funded projects,
and holds four US patents in wireless Internet and
data networking. He is a recipient of the Best Paper Awards in ACM MobiCom’99, ACM MSWiM2000, and ACM/IEEE PADS’97. Dr. Das is an editor of four journals including Computer Networks,
and the Subject Area Editor of mobile computing for
the Journal of Parallel and Distributed Computing.
He has guest-edited special issues of WINET, JPDC,
IEEE PCS, and IEEE Transactions on Computers.
He served as General Chair of WoWMoM-2000 and
2001, WNMC-2001 and MASCOTS’98 and 2002;
General Vice-Chair of MobiCom 2000, HiPC 2000
and 2001; Founder of WoWMoM; TPC Chair of
WoWMoM’98 and WoWMoM’99; Program ViceChair of HiPC’99; and TPC member of numerous
conferences including INFOCOM, MobiCom, ICPP
and IPDPS. He currently serves on IEEE TCPP Executive Committee.
E-mail: [email protected]
Damla Turgut received both her B.S. and M.S. degrees in computer science and engineering from the
University of Texas at Arlington (UTA) in 1994 and
1996, respectively. Currently, she is a Ph.D. candidate and an Assistant Instructor in the Department of
Computer Science and Engineering at UTA. She coauthored a book “Introduction to Computer Science
and Programming with C”. She is a member of the
Center for Research in Wireless Mobility and Networking (CReWMaN). Her current research interests
include mobile ad hoc networks, wireless networks,
mobile computing, mobile and object-oriented databases. She has published more than a dozen research papers in these areas. During 1997–1998,
she was a project leader in Computer Based Training program at the Center for Advanced Engineering and Systems Automation Research (CAESAR).
She served as an officer in a various student organizations. During 1998–1999, she received an outstanding Graduate Student Recognition Committee
Chair and Meritorius Service awards from Graduate
Student Council at UTA. She has been the recipient
of the Texas Telecommunication Engineering Consortium (TxTEC) fellowship since 1999. She is a
member of Upsilon Pi Epsilon (UPE) Honor Society
for Computing Sciences.
E-mail: [email protected]